Topology General Exam

August 15, 2022

Instructions: This is a four hour exam. Your solutions should be legible and clearly organized, written in complete sentences in good mathematical style on your own paper. All work should be your own-no outside sources are permitted-using methods and results from the first year topology course topics. Each problem is worth the same number of points.

Problem 1

Let $\gamma: \mathbb{R} \rightarrow \mathbb{R}^{n}$ be a smooth curve. Prove that the set of real numbers

$$K=\left\{r>0 \mid \text { the sphere of radius } r \text { around } 0 \in \mathbb{R}^{n} \text { is tangent to the image of } \gamma\right\}$$

has measure zero in $\mathbb{R}$.

Problem 2

(a) Let $M$ and $N$ be connected compact smooth manifolds of the same dimension (without boundary), and $f: M \rightarrow N$ a submersion. Prove that $f$ is a covering map.
(b) Suppose $M$ is a connected closed surface ( 2 -manifold), and $f: M \rightarrow S^{2}$ a submersion. Prove that $f$ must, in fact, be a diffeomorphism.

Problem 3

Consider the subset $S$ of $\mathbb{R}^{4}$ defined by the two equations

$$\begin{array}{r} x^{2}+y^{2}-z^{2}-w^{2}=1 \\ x y+z+w=3 \end{array}$$

Prove that $S$ is a smooth manifold and find its dimension.

Problem 4

Consider the smooth, closed curve $\gamma$ in $\mathbb{R}^{3}$ given in cylindrical coordinates ${ }^{1}$ by

$$\gamma(t)=(r(t), \theta(t), z(t)), \quad \begin{aligned} & r(t)=2+\sin t \\ & \theta(t)=2 t \\ & z(t)=\cos t \end{aligned}$$

where $t \in[0,2 \pi]$. Visually, $\gamma$ can be pictured as the boundary of a Möbius band $M \subset \mathbb{R}^{3}-\{r=0\}$ pictured below. Let $\alpha$ be the 1 -form on $\mathbb{R}^{3}-\{r=0\}$ defined by $\alpha=d \theta$.
a) Find $\int_{\gamma} \alpha$.
b) Use your answer to (a) to prove that Möbius band $M$ is not orientable, and in fact $\gamma$ is not the boundary of any smooth, compact, orientable surface in $\mathbb{R}^{3}-\{r=0\}$.

Problem 5

(a) Show that if $f: S^{4} \rightarrow S^{4}$ is continuous, then $f \circ f: S^{4} \rightarrow S^{4}$ must have a fixed point.
(b) By contrast, show that there is a continuous map $f: S^{3} \rightarrow S^{3}$ such that $f \circ f$ has no fixed point.

Problem 6

Let $X_{n}$ denote the $n$-skeleton of a CW complex $X$.
a) Complete the definition: The cellular chain complex $\left(C_{*}^{C W}, d_{*}\right)$ of $X$ is defined by letting $C_{n}^{C W}(X)=H_{n}\left(X_{n}, X_{n-1}\right)$ and then letting $d_{n}: C_{n+1}^{C W}(X) \rightarrow C_{n}^{C W}(X)$ be [you complete the definition].
b) Using your definition from part (a), show that $d_{n-1} \circ d_{n}=0$ for all $n \geq 1$.
c) Prove that $C_{n}^{C W}(X)$ is isomorphic to a free abelian group with one generator for each $n$-cell of $X$.

Problem 7

Let $S^{3} \xrightarrow{i_{1}} S^{3} \vee S^{3} \xrightarrow{i_{2}} S^{3}$ be the two inclusion maps into the wedge. Say that a map $f: S^{3} \vee S^{3} \rightarrow S^{3}$ has type ( $m, n$ ) if the degree of $f \circ i_{1}$ is $m$ and the degree of $f \circ i_{2}$ is $n$. Let $X_{f}=S^{3} \cup_{f}\left(D^{4} \vee D^{4}\right)$, i.e. $X_{f}$ is the pushout

Compute the homology groups of $X_{f}$ if $f$ has type $(9,6)$, describing the homology groups as direct sums of cyclic groups, as usual.

Problem 8

A group $G$ is perfect if its commutator subgroup ${ }^{2}[G, G]$ is all of $G$. The famous Poincaré ’sphere’ is a 3-dimensional manifold $M$ whose fundamental group is a perfect group of order 120. Show that any continuous function $M \rightarrow \mathbb{R} P^{4}$ must be null homotopic.

(You may assume that $f$ is smooth if desired.)